SC · Fourier

Introduction to Bode Plots

A Bode plot draws magnitude in decibels and frequency on a log axis. Then the curved frequency response simplifies into straight-line asymptotes that bend at a corner. Move the corner frequency and see how the straight lines follow.

Land the bend on the target corner

The top is the Bode magnitude plot, with magnitude in decibels. The blue curve is the true response; the gold lines are the asymptotes. Below is the phase plot. Move the corner-frequency slider to land the bend on the dashed target. On the log axis, above the corner you see a straight line of −20 dB per decade.

magnitude (dB)asymptotes

phase (deg)

corner frequency ωc1.00

Corner and asymptote

ωc = 1.00   −20 dB / dec

The bend is far from the target corner. Below the corner it is flat at 0 dB; above it the line drops at −20 dB per decade. The log axis straightens that rolloff into a line.

Corner off

Decibels and the log axis

A Bode plot draws magnitude as 20 log₁₀|H| in decibels and frequency on a log axis. With logs, products become sums, so the decibel magnitudes of several stages in series simply add. And a wide frequency range, say 0.1 to 1000, fits evenly in one picture. A multiplicative, wide-ranging response moves into a world of addition and straight lines.

ObservedB = 20 log₁₀|H|

Decibels are 20 log₁₀|H|.

Choose1st order slope = ?

A first-order slope is −20 dB/decade.

Fill inat ωc: |H| = ?

At the corner the true magnitude is −3 dB.

On your ownat ωc: phase = ?

At the corner the phase is −45 degrees.

The curve becomes straight lines

Drawing a first-order lowpass magnitude in dB on a log axis straightens it into two lines: below the corner ωc a horizontal line at 0 dB, above it a line dropping 20 dB every time the frequency tenfolds — −20 dB per decade. The two lines meet at the corner, where the true curve sits just 3 dB below. So knowing a few corner frequencies lets you sketch the whole response with straight segments. A second-order system doubles the slope to −40 dB per decade.

Read the phase too

A Bode plot is a pair: magnitude and phase. For a first-order lowpass the phase is 0 degrees well below the corner, −45 degrees at the corner, and −90 degrees well above, sliding down smoothly over about two decades around the corner. The magnitude’s corner and the phase’s −45 degrees occur at the same ωc. How much a system attenuates and how much it delays each frequency is read at a glance from these two straight-line plots. The next unit reads bandwidth and filter type straight off this magnitude plot.

Back to the first screen

Each time you moved the corner frequency, the bend slid along the log axis. Below the corner it was always flat 0 dB; above, always a straight −20 dB/decade line; and the two met at the corner. The true curve sat just 3 dB below there. The phase swung from 0 to −90 degrees around the same corner. The power of the Bode plot is to turn a curved frequency response into a few corners and straight segments, so a wide frequency range can be sketched by hand.

A Bode plot draws magnitude as decibels 20 log₁₀|H(ω)| and phase in degrees, both over a log frequency axis. The response straightens into straight-line asymptotes. A first-order lowpass is 0 dB below the corner ωc and −20 dB/decade above, with the true value −3 dB at the corner. Phase goes 0° → −45° at the corner → −90°. The log axis turns products into sums so series stages add in dB, and you sketch the whole from a few corner frequencies. Order n has slope −20n dB/decade.

On to the next unit

Now that you can read a Bode magnitude plot, it is time to pull practical numbers from it. The next unit, bandwidth and filtering, reads the width of the passband — the bandwidth out to −3 dB — straight from the corner, and distinguishes the four filters (lowpass, highpass, bandpass, bandstop) by the shape of the magnitude plot. Multiply a signal’s spectrum by the filter’s magnitude plot and you fix which frequencies survive. The convolution-is-multiplication property of C4 becomes the practical tool of filter design here.